Features

Microstructure parameterization

• flowstress $g$

Kinetics

Deformation

In accordance with the Peirce, Asaro, & Needleman (1983) law, the (average) shear rate is formulated as a power-law kinetic equation \begin{equation} \label{eq: shear rate} \dot\gamma = \dot\gamma_0 \left( \frac{\sqrt{3 J_2}}{M\,g} \right)^n = \dot\gamma_0 \left( \sqrt{\frac{3}{2}}\,\frac{\| \tnsr S^* \|}{M\,g} \right)^n, \end{equation} with $\dot\gamma_0$ a reference shear rate, $n$ the stress exponent (at constant structure), and $M$ an orientation (Taylor) factor.

Structure

Again following the hardening behavior suggested in Peirce, Asaro, & Needleman (1983) the flow stress $g$ evolves in time due to deformation from its initial value $g_0$ towards a saturation value $g_\infty$ according to \begin{align} \label{eq: hardening} \dot g &= \dot\gamma \left( h_0 + h_\text s \ln{\dot\gamma} \right) \left| 1 - g/g_\infty \right|^a \operatorname{sign} \left( 1 - g/g_\infty \right), \end{align} with free parameters $h_0$ and $a$. The parameter $h_\text s = \mathrm d h_0/\mathrm d \ln \dot\gamma$ introduces a strain rate sensitivity of the hardening slope.

To capture more than the power-law rate dependency of the saturation stress inherent in \eqref{eq: shear rate}, we make use of the empirical relation \begin{align} \dot\gamma &= A\left( \sinh \left( B\, g_\infty^\ast\right)^C \right)^D \nonumber \\ &= A\left( \sinh \left( B\, g_\infty\left(\dot\gamma/\dot\gamma_0\right)^{1/n}\right)^C \right)^D, \end{align} where the factor $\left(\dot\gamma/\dot\gamma_0\right)^{1/n}$ corrects the (experimentally observed) saturation stress $g_\infty^\ast$ for the rate sensitivity introduced by the deformation kinetics \eqref{eq: shear rate}. Parameters $A$, $B$, $C$, and $D$ allow for fitting.

The value of $A$ is used to switch between constant saturation stress and rate-sensitive saturation behavior: \begin{equation} g_\infty = \begin{cases} \tau_\text{sat} & \text{if } A = 0 \\ \tau_\text{sat} + \left.\left( \operatorname{asinh} \left( \left( \dot\gamma/A \right)^{1/D} \right) \right)^{1/C} \middle/ \left(B \left( \dot\gamma/\dot\gamma_0 \right )^{1/n}\right)\right. & \text{otherwise} \end{cases} \end{equation}

Kinematics

Plastic velocity gradient

\begin{align} \tnsr L_\text p = \frac{\dot\gamma}{M}\, \frac{\tnsr S^*}{\| \tnsr S^*\|} &= \underbrace{\frac{\dot\gamma_0}{M} \left( \sqrt{\frac{3}{2}}\frac{1}{M\, g}\right)^n}_{k} \, \tnsr S^*\, \| \tnsr S^*\|^{n-1} \end{align} where $k$ abbreviates the constant prefactor in preparation of the next section.

Tangent $\partial\tnsr L_\text p / \partial \tnsr S$

\begin{align*} \tnsr L_\text p,_{\scriptstyle\tnsr S} & = \tnsr L_\text p,_{\scriptstyle\tnsr S^*} \lcontract \,\tnsr S^*,_{\scriptstyle\tnsr S} \\ & = \tnsr L_\text p,_{\scriptstyle\tnsr S^*} \lcontract \underbrace{\left[ \tnsr I \otimes \tnsr I - \frac{1}{3} \tnsr I \odot \tnsr I \right]}_{\tnsrfour Q} \\ & = k \left[\tnsr S^* \|\tnsr S^*\|^{n-1} \right],_{\scriptstyle\tnsr S^*} \lcontract \,\tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \left[{{\| \tnsr S^* \|}^{n-1}} \right],_{\scriptstyle\tnsr S^*} + \| \tnsr S^* \|^{n-1} \, \tnsr S^*,_{\scriptstyle\tnsr S^*} \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot ( n-1 ) \|\tnsr S^*\|^{n-2} \, \frac{\tnsr S^*}{\| \tnsr S^* \|} + \| \tnsr S^*\|^{n-1} \, \tnsr I \otimes \tnsr I \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \tnsr S^* ( n-1 ) \|\tnsr S^*\|^{n-3} + \| \tnsr S^* \|^{n-1} \, \tnsr I \otimes \tnsr I \right] \lcontract \, \tnsrfour Q \\ & = k \left[ \tnsr S^* \odot \tnsr S^* ( n-1 ) \|\tnsr S^*\|^{n-3} \right] \lcontract \tnsrfour Q + k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k ( n-1 ) \, \|\tnsr S^*\|^{n-3} \, \left[ \tnsr S^* \odot \tnsr S^* \lcontract \, \tnsr I \otimes \tnsr I - \frac{1}{3}\, \tnsr S^* \odot \tnsr S^* \lcontract \, \tnsr I \odot \tnsr I \right] \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k \left( n-1 \right) \, \|\tnsr S^*\|^{n-3} \, \left[ \tnsr S^* \odot \tnsr S^* - \frac{1}{3} \underbrace{(\tnsr S^* : \tnsr I)}_{=\,\operatorname{tr}(\tnsr S^*)\,=\,\tnsr 0} \tnsr S^* \odot \tnsr I \right] \\ & = k \, \| \tnsr S^* \|^{n-1} \, \tnsrfour Q + k \left( n-1 \right) \, \|\tnsr S^*\|^{n-3} \, \tnsr S^* \odot \tnsr S^* \\ & = k \, \| \tnsr S^* \|^{n-1} \left[ \tnsrfour Q + ( n-1 ) \, \frac{\tnsr S^*}{\| \tnsr S^* \|} \odot \frac{\tnsr S^*}{\| \tnsr S^* \|} \right] \end{align*} It is useful to rewrite this equation in index notation. \begin{align*} & \vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j \ \frac{\partial \{L_\text p\}_{ij}}{\partial S_{kl}} = \\ & \vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right] \end{align*}

A change of tensor basis is performed to transform $\left [\tnsr L_\text p,_{\scriptstyle\tnsr S}\right]^\text R = \tnsr L_\text p;_{\scriptstyle\tnsr S}$. In index notation, this corresponds to: \begin{align*} \Big[\vctr g^i \otimes \vctr g_k \otimes \vctr g_l \otimes \vctr g^j & \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right]\Big]^\text R = \\ \phantom{\Big[}\vctr g^i \otimes \vctr g^j \otimes \vctr g_k \otimes \vctr g_l & \ k \, \| \tnsr S^* \|^{n-1} \left[ \delta_i^k\,\delta^l_j - 1/3\; \delta_{ij}\,\delta^{kl} + (n-1)\| \tnsr S^* \|^{-2} \, {S^*}_{ij}\, {S^*}^{kl} \right] = \tnsr L_\text p;_{\scriptstyle\tnsr S} \end{align*}

Parameters in material configuration

Appendix

Norm of a tensor

Deviatoric part of a tensor

Derivative of the norm of a tensor with respect to that tensor

Derivative of the deviator of a tensor with respect to that tensor

References

[1]

D. Peirce, R. Asaro, & A. Needleman
Material rate dependence and localized deformation in crystalline solids
Acta Metall. 31 (1983) 1951–1976